US President Donald Trump’s lawyer, Rudy Giuliani, has said that Robert Mueller’s investigation into whether Trump’s 2016 election campaign colluded with Russia will end by September 1, prompting a rebuke from within the Justice Department.

Giuliani told The New York Times on Sunday that the office of Robert Mueller – the special counsel who is leading the investigation – had shared the timeline some two weeks ago.

Trump, who has repeatedly described the investigation as a “witch hunt,” views it as a stain on his presidency and wants it over – an end Giuliani apparently sought to further through his comments on the probe’s conclusion.

“You don’t want another repeat of the 2016 election where you get contrary reports at the end and you don’t know how it affected the election,” he told the Times, referring to the investigation’s possible impact on midterm elections in November.

However, a source familiar with the probe called the September 1 deadline “entirely made-up” and “another apparent effort to pressure the special counsel to hasten the end of his work”.

“He’ll wrap it up when he thinks he’s turned over every rock, and when that is will depend on how cooperative witnesses, persons of interest and maybe even some targets are, if any of those emerge, and on what new evidence he finds, not on some arbitrary, first-of-the-month deadline one of the president’s attorneys cooks up,” said the source, a US official who spoke on condition of anonymity.

Former FBI director James Comey – fired by Trump last year – announced that the bureau would reopen an investigation into Hillary Clinton’s misuse of a private email server just 11 days before the 2016 presidential vote, a move that may have helped cost her the election.

Trump’s former campaign chairman Paul Manafort, his first national security adviser, Michael Flynn, and an assortment of campaign aides have all been caught in Mueller’s investigation, variously pleading guilty, cutting a plea deal or fighting the case in court.

But a year after the probe began, the big questions remain unanswered: Did Trump’s campaign collude with the Kremlin to skew the 2016 election? What did the president know and when did he know it? Did he obstruct justice? Can a sitting president be indicted?